Abstract: Let f be a Morse-Bott function on a compact oriented finite dimensional manifold M. The polynomial Morse inequalities and a perturbation technique approximating any Morse-Bott function by a Morse function show immediately that if MB_t(f) is the Morse-Bott polynomial of f , and P_t(M) is the Poincare polynomial of M , there exists a polynomial R(t) such that MB_t(f) = P_t(M) + (1+t)R(t). We prove that R(t) is a polynomial with non-negative integer coefficients, using the Morse Homology Theorem, and by studying the ranks of the kernels of the differentials of the Morse-Smale-Witten complexes of the Morse functions involved in the approximation scheme. Our method works when all the critical submanifolds are oriented or when Z_2 coefficients are used. Our proof is much more direct than all previous proofs. This is a joint work with David E. Hurtubise.